Sunday, March 31, 2013

Risk Averse Bidders in Auctions

I am very puzzled: in a lecture course on game theory, I heard the lecturer state that "risk averse" bidders will not behave as standard theory says they should: they will bid too high since they are averse to the risk of not winning the auction.

But wait: why is that called risk averse, rather than being averse to the risk of paying too much for an item? This seems arbitrary. Any explanations?

14 comments:

It seems like a misuse of the term... risk aversion has to do with ratio of the first and second derivative of the utility function. It's entirely a preferences issue.

Presumably they're thinking of something like a cognitive bias where people poorly assess risk?

This came up in my gender macro class too on gender differences in risk aversion and savings behavior... some of the literature was good but sometimes the terms were used in a very vague way with implications of irrationality, etc.

In finance work, it was usually defined as aversion to expected price volatility (which was usually thought to be simply past price volatility extrapolated into the future). The standard deviation of past price movements is higher, the asset is riskier.

I guess this can be mapped to your definition above: the higher the second derivative compared to the first, the greater we would expect price volatility to be, right?

Ya, I'd think so. Just the undifferentiated micro use of the term concerns uncertainty in the level of wealth. If that's held as an asset which trades at a volatile price you're dealing with the same thing.

This is all I'm talking about: http://en.wikipedia.org/wiki/Risk_aversion#Absolute_risk_aversion

I completely agree with your sense that this doesn't have anything to do with rationality. I guess the question is, are they goofing an explanation or are they thinking of something fundamentally different.

Suppose that the bidder expects that price to be 5, and he expects to receive a rent of 10 after purchasing it for 5. The "standard theory" would say that he would bid 5 + epsilon. But he might make a bid bigger than that to ensure he gets at least some rent, if less.

OK, bu why is that being "risk averse"? Does he have no opportunity costs here? Is there nothing else he might be bidding on where he could simply pay 5 + ε to get the rent of 10? If there isn't, isn't that a good sign that bidding 5 + ε might be overpaying?

In other words, why is a fear of missing out on a return a sign of being more "risk averse" than a fear of overpaying?

This is why I think you can reasonably frame it as being risk averse. You need to reverse the way you are thinking about it. Conditional on making the bid at all, bidding 5+epsilon gives probability p that he will not win the bid, in which case his expected pay out is (1-p)*(15 - (5+epsilon) + p*(-1*(5+epsilon).

Now suppose that if he bids 5+gamma, such that gamma>episilon, he wins the bid with certainty. Then his expected pay out is (15-(5+gamma)).

Obviously the second scenario has a lower expected mean and lower variance than the first. Someone who is risk neutral would not make that choice. Someone who is risk averse might.

As to the other concerns, those are restrictive assumptions in the model. Whether they are reasonable or not depends on the context (e.g. how liquid the market is). They aren't weird assumptions for game theory.

I'm not trying to be "that guy," but Gene if you explain exactly what the auction rules were, I can tell you definitively whether the lecturer was right or wrong. I can imagine one type of auction where his statement makes perfect sense, but it will be easier if you tell me the exact setup first.

That is nothing to be feared, or at least, nothing other than the waste of everyone's time. If you fear overbidding, it is because you fear being caught up in the moment. Tossing some low bids about does nothing to cure that. The solution is to fix your maximum bid before it begins. The winning bid will be among the few that value it most, but they should reflect on that value soberly prior, rather its value in the moment.

Well, the bid is a gamble with possible values of v-b or 0, where v is your valuation and b is your bid and its expected value is the probability of winning ( a positive function of b) times (v-b). Suppose the risk-neutral bidder would bid x. Suppose you raised the bid so that the probability of winning rose just enough to offset the fall in v-b, so that the expected value of the gamble was unchanged. Since you have the same expected return but a lower variance with this higher bid, a risk-averse person would be better off.

Well, ignore that last post. Of course the expected monetary value of the bet will be lower with a higher bid than what a risk neutral bidder would bid. Let me try again: A risk-averse person will trade off some expected return for lower variance. You could make your gamble zero-risk by bidding your value - but of course you won't be willing to give up that much expected value- i.e., all of it!

But the larger point is that I agree with Gene that his lecturer mis-described what's going on here.

Just for example's sake, suppose we have a private value auction with two risk-neutral bidders and values are private information drawn from a uniform distribution on {0,1}. Then there is a Nash equilibrium where each bids half his valuation. If we make them risk-averse, with utility equal to the square root of income, then the equilibrium has each bidding 2/3 of his valuation. The expected value of the gamble is lower but the variance is lower as well, compared to the risk-neutral case.

"Risk averse" is operationally defined as you having a concave utility function in terms of money. I.e diminishing marginal utility from additional increments in your total # of dollars.

So this makes sense: You wouldn't take a 50/50 wager of $W, because if you start with E dollars right now (for Endowment), then

U(E) > (1/2)U(E-W) + (1/2)U(E+W)

So this working definition gets us what we want: A person wouldn't even take a fair wager if he's risk averse, let alone would he take a losing wager (like playing Blackjack or, worse yet, the lotto).

(If a person were risk neutral, then he would be indifferent between taking a fair wager versus walking away.)

So if I'm risk averse in this sense, and I'm in a standard first-price auction where I know my valuation of the prize for sure, then the only uncertainty is over the bids of everyone else. So you can come up with what some rule that a risk-neutral person would do. He's obviously going to bid less than his valuation amount, but note the tradeoff: the higher his bid, the more likely he will win, but the smaller his net payoff.

So flipping it around, if he decreases his bid, then he is less likely to win but will win a greater amount if he does.

This is the sense in which it's non-arbitrarily "riskier" to bid lower, just like it's riskier to buy a lotto ticket than not. You could philosophically say, "Ah but if I don't buy the lotto ticket, am I not 'risking' the forfeiture of the jackpot?" but that's not right in terms of the economics. It objectively is riskier to buy a lotto ticket than to refrain from buying one.

So back to the auction, if you see how it could objectively be riskier to place a lower bid, then someone who is risk averse will bid higher (in equilibrium) than someone who is risk neutral, other things equal.

Yes, Ryan, Bob, et al., I get it now: this is an artifact of the weird way risk is defined in finance: if I am in a car going over a cliff, it is riskless to stay in the car (my "return" is death with 100% certainty), but "risky" to try to jump out (since now there is some chance I might live, which ups the variance).